$K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint Operator

This paper will generalize $b$-frames, a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product, we will use a new product called the $b$-dual product, which is constructed via a bilinear mapping. We will introduce new results about this product, about $b$-frames and $K$-$b$-frames and we will also give some examples of both $b$-frames and $K$-$b$-frames that have never been given before. We will express the reconstruction formula of the elements of the Hilbert space. We will also study the stability and preservation of both $b$-frames and $K$-$b$-frames and to do so, we will give the equivalent of the adjoint operator according to the $b$-dual product.